# Countability of “center” points of line segments in complement of Cantor set

So, start with the set [0,1] of the real line. Remove the middle third, and keep removing the middle thirds of the remaining line segments as usual when making the Cantor set.

Each time you remove a 3rd, add the middle point of the line segment you just removed to another set, S (initially empty). In the end you have the Cantor set C, and another set, S. Are the points in S countable?

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One obvious way of checking that $S$ is countable is to use that $\mathbb Q$ itself is countable, and to notice that $S\subset \mathbb Q$. – Andrés Caicedo Jan 29 '13 at 4:00

Yes, $S$ is countable. At stage $n$ you removed $2^n$ open intervals, so you added $2^n$ points to $S$. $S$ is therefore the union of countably infinitely many finite sets and is therefore countable.
Another way to prove this is to notice that the open intervals that are removed to form the middle-thirds Cantor set are pairwise disjoint, and each contains exactly one point of $S$. Let $\mathscr{I}$ be the set of removed intervals. For each $s\in S$ let $I_s$ be the member of $\mathscr{I}$ containing $s$, and let $q_s\in I_s\cap\Bbb Q$. Clearly the map from $S\to\Bbb Q:s\mapsto q_s$ is injective (one-to-one), so $|S|\le|\Bbb Q|$, and $S$ is therefore countable.
@GodOfDjinns: Actually, the second and third centres of deleted intervals are $1/6$ and $5/6$. Or are you now mapping the points of $S$ to some different subset of $[0,1]$? – Brian M. Scott Jan 29 '13 at 3:43
@GodOfDjinns: You get precisely the dyadic rationals in $(0,1)$, i.e., the rational numbers that can be expressed in lowest terms in the form $\frac{k}{2^n}$ for some odd positive integer $k$ and positive integer $n$. This is a countable dense subset of $(0,1)$. – Brian M. Scott Jan 29 '13 at 3:59